The very first question for today's lesson is "So what is our maximum volume?". This is in reference to the volume model the students found yesterday. Next I ask "Are you sure?" which leads to a discussion as to how we can be sure whether this is the maximum value (Math Practice 4). It should come out that we can't be sure because we didn't find all the possible volumes.

Now, the students build the algebraic model f(x)=x(8.5 – 2x)(11 – 2x). This is given that the graph paper is 8.5x11. My tactics for building this model will change depending on the independence and confidence of my students. If they are mathematically proficient, I give them an opportunity to play with it on their own with no introductory coaching. Scaffolding can then be provided to individuals who are struggling. If my class is a bit less confident, I may build this more as a guided investigation. My educreations video shows some scaffolding I use in this lesson.

I discourage students from distributing this polynomial expression since it is not necessary, unless they have a burning desire to do so. If it comes up, we talk about how the two forms are equivalent and can be used in place of each other.

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Now that they have an algebraic model, the students graph it on a graphing calculator to find the exact solution (we use TI-84s). My students always ask me why we don’t do it the easy way first (often with exasperation). First, I might (jokingly) tell them it's because I am mean (my favorite reason for everything). Next, I tell them because the easy way often doesn’t really help them understand what is going on. If you just use a short cut without understanding the math behind it, it's not really going to help you after all.

I have them match their screen window with the dimensions of the graph from our scatter plot in the first day of this lesson. I believe this will increase their awareness of the connections between what they did and what is on the screen. Next, they type the function into the Y= and then graph.

The graph they obtain will look extremely similar to the graph they created from their physical models. Now we want to find the exact maximum volume. We start by using the tracing function. Once they have a pretty good idea using trace, I show them how to use MAXIMUM. I then ask them to compare the three methods we have used; the physical model, tracing and then the MAX button. While this answer is obvious, it helps them to practice evaluating the effectiveness of different methods (Math Practice 5).

Now we will look at the entire graph. I introduce this by asking them if they think our graph covers all the possible portions of this function. Bringing up a specific example like finding f(5) helps the students realize that our graph extends beyond our current view.

I decided to give them the window to use. At least in this problem, having a clear and consistent window for all of the students is more important than them figuring out the appropriate window. Here is the window I give them:

Xmin = -1

Xmax = 7

Xscl = 1

Ymin = -20

Ymax = 70

Yscl = 5

Once the students can see the entire graph, we talk about some of the properties highlighting the differences between the entire graph and the portion used in our model (Math Practice 2).

The first things they find are the domain and range. I ask "Why is the domain and range for the whole function so different that the domain and range for the volume model?" and then "What about the second part of the graph in the first quadrant? Doesn’t it play a part of our volume model?" This is a time for them to really analyze the differences between our model and the entire function (Math Practice 7). A think-pair-share is appropriate for these questions.

Now, they find the local minimum (they already know the local maximum). One important aspect to discuss is how to write the local minimum or maximum. I have them identify the full ordered pair and then ask them which number represents the min or max. For example, if the local maximum is at (2, 15) then the maximum is 15. Relating it back to the volume model helps the students make meaning.

Finally, they look at the intervals of increase and decrease. Our original model simply had one interval of increase and one of decrease. This has two of each. This may be hard for some students to see. I have them make a sketch on their paper and use small arrows to show the intervals. Next, they write intervals. They check with their neighbor and once everyone has an idea, we discuss as a class. I may make my own sketch on the white board to model the solution.

The first three problems in this Homework ask the students to find the local minimums and maximums of graphed polynomial functions. They are also asked to find intervals of increase and decrease. Next they are asked some to categorize these three polynomials as having an even/odd degree, multiple congruent zeros or imaginary roots.

The second portion of this homework gives the students a modeling problem involving the creation of a rain gutter (Math Practice 4). This will give them a further opportunity to deepen their ability to model with polynomials. Since not all students have access to a graphing calculator, I chose not to include that portion of this problem. We will extend this problem using the graphing calculator during the chapter review.